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Chapter Contents

Chapter Introduction

NAG Toolbox: nag_quad_withdraw_1d_gauss_wset (d01bb)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_quad_1d_gauss_wset (d01bb) returns the weights and abscissae appropriate to a Gaussian quadrature formula with a specified number of abscissae. The formulae provided are Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite.

Note: this function is scheduled to be withdrawn, please see d01bb in Advice on Replacement Calls for Withdrawn/Superseded Routines..

Syntax

[weight, abscis, ifail] = d01bb(d01xxx, a, b, itype, n)

[weight, abscis, ifail] = nag_quad_withdraw_1d_gauss_wset(d01xxx, a, b, itype, n)

Description

nag_quad_1d_gauss_wset (d01bb) returns the weights and abscissae for use in the Gaussian quadrature of a function

f (x)

. The quadrature takes the form

S = \sum_{i = 1}^{n} w_{i} f (x_{i})

where

w_{i}

are the weights and

x_{i}

are the abscissae (see Davis and Rabinowitz (1975), Fröberg (1970), Ralston (1965) or Stroud and Secrest (1966)).

Weights and abscissae are available for Gauss–Legendre, rational Gauss, Gauss–Laguerre and Gauss–Hermite quadrature, and for a selection of values of

n

(see Arguments).

(a)

Gauss–Legendre Quadrature:

S ≃ \int_{a}^{b} f (x) d x

where

a

and

b

are finite and it will be exact for any function of the form

f (x) = \sum_{i = 0}^{2 n - 1} c_{i} x^{i} .

(b)

Rational Gauss quadrature, adjusted weights:

S ≃ \int_{a}^{\infty} f (x) d x (a + b > 0) or S ≃ \int_{- \infty}^{a} f (x) d x (a + b < 0)

and will be exact for any function of the form

f (x) = \sum_{i = 2}^{2 n + 1} \frac{c_{i}}{{(x + b)}^{i}} = \frac{\sum_{i = 0}^{2 n - 1} c_{2 n + 1 - i} {(x + b)}^{i}}{{(x + b)}^{2 n + 1}} .

(c)

Gauss–Laguerre quadrature, adjusted weights:

S ≃ \int_{a}^{\infty} f (x) d x (b > 0) or S ≃ \int_{- \infty}^{a} f (x) d x (b < 0)

and will be exact for any function of the form

f (x) = e^{- b x} \sum_{i = 0}^{2 n - 1} c_{i} x^{i} .

(d)

Gauss–Hermite quadrature, adjusted weights:

S ≃ \int_{- \infty}^{+ \infty} f (x) d x

and will be exact for any function of the form

f (x) = e^{- b {(x - a)}^{2}} \sum_{i = 0}^{2 n - 1} c_{i} x^{i} (b > 0) .

(e)

Gauss–Laguerre quadrature, normal weights:

S ≃ \int_{a}^{\infty} e^{- b x} f (x) d x (b > 0) or S ≃ \int_{- \infty}^{a} e^{- b x} f (x) d x (b < 0)

and will be exact for any function of the form

f (x) = \sum_{i = 0}^{2 n - 1} c_{i} x^{i} .

(f)

Gauss–Hermite quadrature, normal weights:

S ≃ \int_{- \infty}^{+ \infty} e^{- b {(x - a)}^{2}} f (x) d x

and will be exact for any function of the form

f (x) = \sum_{i = 0}^{2 n - 1} c_{i} x^{i} .

Note: the Gauss–Legendre abscissae, with

a = - 1

,

b = + 1

, are the zeros of the Legendre polynomials; the Gauss–Laguerre abscissae, with

a = 0

,

b = 1

, are the zeros of the Laguerre polynomials; and the Gauss–Hermite abscissae, with

a = 0

,

b = 1

, are the zeros of the Hermite polynomials.

References

Davis P J and Rabinowitz P (1975) Methods of Numerical Integration Academic Press

Fröberg C E (1970) Introduction to Numerical Analysis Addison–Wesley

Ralston A (1965) A First Course in Numerical Analysis pp. 87–90 McGraw–Hill

Stroud A H and Secrest D (1966) Gaussian Quadrature Formulas Prentice–Hall

Parameters

Compulsory Input Parameters

1: – string

String specifying the quadrature formula to be used:

'd01baz', for Gauss–Legendre weights and abscissae;
'd01bay', for rational Gauss weights and abscissae;
'd01bax', for Gauss–Laguerre weights and abscissae;
'd01baw', for Gauss–Hermite weights and abscissae.

2: $a$ – double scalar

3: $b$ – double scalar

The quantities

a

and

b

as described in the appropriate sub-section of Description.

4: $itype$ – int64int32nag_int scalar

Indicates the type of weights for Gauss–Laguerre or Gauss–Hermite quadrature (see Description).

$itype = 1$: Adjusted weights will be returned.
$itype = 0$: Normal weights will be returned.

Constraint:

itype = 0

or

1

.

For Gauss–Legendre or rational Gauss quadrature, this argument is not used.

5: $n$ – int64int32nag_int scalar

n

, the number of weights and abscissae to be returned.

Constraint:

n = 1

,

2

,

3

,

4

,

5

,

6

,

8

,

10

,

12

,

14

,

16

,

20

,

24

,

32

,

48

or

64

.

Optional Input Parameters

None.

Output Parameters

1: $weight (n)$ – double array: The n weights. For Gauss–Laguerre and Gauss–Hermite quadrature, these will be the adjusted weights if $itype = 1$ , and the normal weights if $itype = 0$ .
2: $abscis (n)$ – double array: The n abscissae.
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W $ifail = 1$: The N-point rule is not among those stored. If the soft fail option is used, the weights and abscissae returned will be those for the largest valid value of n less than the requested value, and the excess elements of weight and abscis (i.e., up to the requested n) will be filled with zeros.

$ifail = 2$: The value of a and/or b is invalid.

Rational Gauss: $a + b = 0.0$
Gauss–Laguerre: $b = 0.0$
Gauss–Hermite: $b \leq 0.0$

If the soft fail option is used the weights and abscissae are returned as zero.

W $ifail = 3$: Laguerre and Hermite normal weights only: underflow is occurring in evaluating one or more of the normal weights. If the soft fail option is used, the underflowing weights are returned as zero. A smaller value of n must be used; or adjusted weights should be used ( $itype = 1$ ). In the latter case, take care that underflow does not occur when evaluating the integrand appropriate for adjusted weights.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The weights and abscissae are stored for standard values of a and b to full machine accuracy.

Further Comments

Timing is negligible.

Example

This example returns the abscissae and (adjusted) weights for the six-point Gauss–Laguerre formula.

Open in the MATLAB editor: d01bb_example

function d01bb_example


fprintf('d01bb example results\n\n');

a = 0;
b = 1;
itype = int64(1);
n = int64(6);

[weight, abscis, ifail] = ...
  d01bb( ...
	 'd01bax', a, b, itype, n);

fprintf('  Weights    Abscissae\n');
fprintf('%9.4f%12.4f\n',[weight abscis]');

d01bb example results

  Weights    Abscissae
   0.5735      0.2228
   1.3693      1.1889
   2.2607      2.9927
   3.3505      5.7751
   4.8868      9.8375
   7.8490     15.9829

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

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